Predictors of frog survival following translocations - analyses
Roland Knapp
The goal of the analyses in this notebook is to identify predictors of frog survival following translocation. The analyses are conducted in a Bayesian framework using the R package rstanarm.
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arlibrary(rstanarm)This is rstanarm version 2.21.3
- See https://mc-stan.org/rstanarm/articles/priors for changes to default priors!
- Default priors may change, so it's safest to specify priors, even if equivalent to the defaults.
- For execution on a local, multicore CPU with excess RAM we recommend calling
options(mc.cores = parallel::detectCores())
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dstudent_t, pstudent_t, qstudent_t, rstudent_tlibrary(loo)This is loo version 2.5.1
- Online documentation and vignettes at mc-stan.org/loo
- As of v2.0.0 loo defaults to 1 core but we recommend using as many as possible. Use the 'cores' argument or set options(mc.cores = NUM_CORES) for an entire session. library(broom.mixed)
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chisq.test, fisher.testsource("classification_summary.R")Read in frog translocation dataset
- Read in data, create variables for analysis, standardize continuous variables.
- arm package used to rescale continuous variables (substract mean, divide by 2 SD), manual centering of binary variables.
d1 <- read_csv(here::here("data", "clean", "frog_translocation_final.csv")) %>%
drop_na(bd_load) %>% # drop xx records from 70xxx where bd_load is NA
mutate(
first = if_else(order == 1, 1, 0),
surv = if_else(survival < 0.5, 0, 1),
male = if_else(sex == "m", 1, 0),
elev_lo = if_else(elevation < 3020, 1, 0), # using interval defined by cut_interval, n = 2
elev_z = arm::rescale(elevation),
snowt_z = arm::rescale(snow_t),
snowt1_z = arm::rescale(snow_t1),
day_z = arm::rescale(day),
length_z = arm::rescale(length),
bdload_l = log10(bd_load + 1),
bdload_z = arm::rescale(bdload_l),
shore_c = shore - mean(shore),
male_c = male - mean(male),
elevlo_c = elev_lo - mean(elev_lo),
first_c = first - mean(first),
across(c(shore_c, male_c, elevlo_c, first_c), round, 3),
across(c(site_id, shore_c, first_c, donor, male_c, elevlo_c, year), as.factor),
surv = as.integer(surv))Rows: 779 Columns: 16── Column specification ────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────
Delimiter: ","
chr (2): pit_tag_ref, sex
dbl (13): site_id, elevation, shore, snow_t, snow_t1, year, day, order, donor, length, weight, bd_load, survival
date (1): date
ℹ Use `spec()` to retrieve the full column specification for this data.
ℹ Specify the column types or set `show_col_types = FALSE` to quiet this message.# Add a descriptive translocation_id
d1 <- d1 %>%
distinct(site_id, year) %>%
arrange(site_id, year) %>%
group_by(site_id) %>%
mutate(transno = seq_len(n())) %>%
ungroup(site_id) %>%
unite(translocation_id, c("site_id", "transno"), remove = FALSE) %>%
select(-transno) %>%
mutate(translocation_id = as.factor(translocation_id)) %>%
inner_join(d1, by = c("site_id", "year")) %>%
relocate(translocation_id, .after = site_id)Null values - check
d1 %>% summarize(across(everything(), ~ sum(is.na(.))))Model using non-standardized predictors
Model specification
m1a <- stan_glm(
surv ~ length + snow_t + snow_t1 + day + bdload_l + elevation + first + male + donor,
data = d1,
family = binomial,
chains = 4,
iter = 5000*2,
cores = 4,
seed = 84735)starting worker pid=10413 on localhost:11976 at 13:34:41.032
starting worker pid=10442 on localhost:11976 at 13:34:41.153
starting worker pid=10471 on localhost:11976 at 13:34:41.276
starting worker pid=10500 on localhost:11976 at 13:34:41.395
SAMPLING FOR MODEL 'bernoulli' NOW (CHAIN 1).
Chain 1:
Chain 1: Gradient evaluation took 7.5e-05 seconds
Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 0.75 seconds.
Chain 1: Adjust your expectations accordingly!
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SAMPLING FOR MODEL 'bernoulli' NOW (CHAIN 2).
Chain 2:
Chain 2: Gradient evaluation took 2.9e-05 seconds
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Chain 2: Adjust your expectations accordingly!
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SAMPLING FOR MODEL 'bernoulli' NOW (CHAIN 3).
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SAMPLING FOR MODEL 'bernoulli' NOW (CHAIN 4).
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Chain 4: Gradient evaluation took 2.8e-05 seconds
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Chain 4: Priors & chain diagnostics
prior_summary(m1a)Priors for model 'm1a'
------
Intercept (after predictors centered)
~ normal(location = 0, scale = 2.5)
Coefficients
Specified prior:
~ normal(location = [0,0,0,...], scale = [2.5,2.5,2.5,...])
Adjusted prior:
~ normal(location = [0,0,0,...], scale = [0.526,0.049,0.073,...])
------
See help('prior_summary.stanreg') for more detailsmcmc_trace(m1a, size = 0.1)
mcmc_dens_overlay(m1a)
summary(m1a)
Model Info:
function: stan_glm
family: binomial [logit]
formula: surv ~ length + snow_t + snow_t1 + day + bdload_l + elevation +
first + male + donor
algorithm: sampling
sample: 20000 (posterior sample size)
priors: see help('prior_summary')
observations: 767
predictors: 11
Estimates:
mean sd 10% 50% 90%
(Intercept) -8.5 2.3 -11.5 -8.5 -5.5
length 0.0 0.0 0.0 0.0 0.0
snow_t 0.0 0.0 0.0 0.0 0.0
snow_t1 0.0 0.0 0.0 0.0 0.0
day 0.0 0.0 0.0 0.0 0.0
bdload_l -0.1 0.1 -0.1 -0.1 0.0
elevation 0.0 0.0 0.0 0.0 0.0
first 0.4 0.2 0.1 0.4 0.7
male 0.3 0.2 0.1 0.3 0.5
donor70567 -1.3 0.4 -1.9 -1.3 -0.8
donor72996 -2.2 0.4 -2.7 -2.2 -1.7
Fit Diagnostics:
mean sd 10% 50% 90%
mean_PPD 0.3 0.0 0.3 0.3 0.3
The mean_ppd is the sample average posterior predictive distribution of the outcome variable (for details see help('summary.stanreg')).
MCMC diagnostics
mcse Rhat n_eff
(Intercept) 0.0 1.0 24916
length 0.0 1.0 25653
snow_t 0.0 1.0 22266
snow_t1 0.0 1.0 21140
day 0.0 1.0 22645
bdload_l 0.0 1.0 29642
elevation 0.0 1.0 24654
first 0.0 1.0 23563
male 0.0 1.0 28297
donor70567 0.0 1.0 20512
donor72996 0.0 1.0 18851
mean_PPD 0.0 1.0 23850
log-posterior 0.0 1.0 8254
For each parameter, mcse is Monte Carlo standard error, n_eff is a crude measure of effective sample size, and Rhat is the potential scale reduction factor on split chains (at convergence Rhat=1).- mcmc diagnostics all look good.
List model variable names
get_variables(m1a) [1] "(Intercept)" "length" "snow_t" "snow_t1" "day" "bdload_l" "elevation" "first"
[9] "male" "donor70567" "donor72996" "accept_stat__" "stepsize__" "treedepth__" "n_leapfrog__" "divergent__"
[17] "energy__" Posterior predictive check
pp_check(m1a, plotfun = "stat") +
xlab("Frog survival rate")
- Posterior predictive distribution implied by regression model closely matches original data (based on mean frog survival).
Calculate leave-one-out cross-validation (loo)
loo_m1a <- loo(m1a, cores = 4, save_psis = TRUE)
print(loo_m1a)
Computed from 20000 by 767 log-likelihood matrix------
Monte Carlo SE of elpd_loo is 0.0.
All Pareto k estimates are good (k < 0.5).
See help('pareto-k-diagnostic') for details.Posterior analysis
tidy(m1a, conf.int = TRUE, conf.level = 0.90)
mcmc_intervals(m1a, point_est = "median", prob = 0.80, prob_outer = 0.9) +
geom_vline(xintercept = 0, color = "red", linetype = "dashed", lwd = 0.5) + # add 0-line to accentuate default line
ggtitle("Model with non-standardized predictors only",
"Posterior distributions with medians & 90% uncertainty intervals")
mcmc_areas(m1a, area_method = "equal height", prob = 0.90) +
geom_vline(xintercept = 0, color = "red", linetype = "dashed", lwd = 0.5) +
ggtitle("Model with non-standardized predictors only",
"Posterior distributions with medians & 90% uncertainty intervals")
mean(bayes_R2(m1a))[1] 0.2728229- Using non-standardized predictors makes plot hard to read due to
vastly different scales of predictors.
- Based on 90% uncertainty intervals, important predictors are snow_t1, elevation, first, donor70567, and donor72996.
Model using standardized predictors
Model specification
m1b <- stan_glm(
surv ~ length_z + snowt_z + snowt1_z + day_z + bdload_z + elev_z + first_c + male_c + donor,
data = d1,
family = binomial,
chains = 4,
iter = 5000*2,
cores = 4,
seed = 84735)starting worker pid=10862 on localhost:11976 at 13:35:10.488
starting worker pid=10891 on localhost:11976 at 13:35:10.614
starting worker pid=10920 on localhost:11976 at 13:35:10.733
starting worker pid=10949 on localhost:11976 at 13:35:10.863
SAMPLING FOR MODEL 'bernoulli' NOW (CHAIN 1).
Chain 1:
Chain 1: Gradient evaluation took 4.8e-05 seconds
Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 0.48 seconds.
Chain 1: Adjust your expectations accordingly!
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SAMPLING FOR MODEL 'bernoulli' NOW (CHAIN 2).
Chain 2:
Chain 2: Gradient evaluation took 2.7e-05 seconds
Chain 2: 1000 transitions using 10 leapfrog steps per transition would take 0.27 seconds.
Chain 2: Adjust your expectations accordingly!
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SAMPLING FOR MODEL 'bernoulli' NOW (CHAIN 3).
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Chain 3: Gradient evaluation took 2.6e-05 seconds
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Chain 4: Gradient evaluation took 3.2e-05 seconds
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Chain 4: Priors, chain diagnostics, posterior predictive check
prior_summary(m1b)Priors for model 'm1b'
------
Intercept (after predictors centered)
~ normal(location = 0, scale = 2.5)
Coefficients
Specified prior:
~ normal(location = [0,0,0,...], scale = [2.5,2.5,2.5,...])
Adjusted prior:
~ normal(location = [0,0,0,...], scale = [5.00,5.00,5.00,...])
------
See help('prior_summary.stanreg') for more detailsmcmc_trace(m1b, size = 0.1)
mcmc_dens_overlay(m1a)
summary(m1b)
Model Info:
function: stan_glm
family: binomial [logit]
formula: surv ~ length_z + snowt_z + snowt1_z + day_z + bdload_z + elev_z +
first_c + male_c + donor
algorithm: sampling
sample: 20000 (posterior sample size)
priors: see help('prior_summary')
observations: 767
predictors: 11
Estimates:
mean sd 10% 50% 90%
(Intercept) 0.4 0.4 0.0 0.4 0.9
length_z 0.0 0.2 -0.3 0.0 0.3
snowt_z 0.0 0.2 -0.3 0.0 0.3
snowt1_z -1.3 0.2 -1.6 -1.3 -1.0
day_z 0.2 0.2 -0.1 0.2 0.4
bdload_z -0.2 0.2 -0.4 -0.2 0.1
elev_z 1.5 0.3 1.2 1.5 1.9
first_c0.353 0.4 0.2 0.1 0.4 0.7
male_c0.548 0.3 0.2 0.1 0.3 0.5
donor70567 -1.3 0.4 -1.9 -1.3 -0.8
donor72996 -2.2 0.4 -2.7 -2.2 -1.7
Fit Diagnostics:
mean sd 10% 50% 90%
mean_PPD 0.3 0.0 0.3 0.3 0.3
The mean_ppd is the sample average posterior predictive distribution of the outcome variable (for details see help('summary.stanreg')).
MCMC diagnostics
mcse Rhat n_eff
(Intercept) 0.0 1.0 21959
length_z 0.0 1.0 25653
snowt_z 0.0 1.0 22266
snowt1_z 0.0 1.0 21140
day_z 0.0 1.0 22645
bdload_z 0.0 1.0 29642
elev_z 0.0 1.0 24654
first_c0.353 0.0 1.0 23563
male_c0.548 0.0 1.0 28297
donor70567 0.0 1.0 20512
donor72996 0.0 1.0 18851
mean_PPD 0.0 1.0 23850
log-posterior 0.0 1.0 8254
For each parameter, mcse is Monte Carlo standard error, n_eff is a crude measure of effective sample size, and Rhat is the potential scale reduction factor on split chains (at convergence Rhat=1).- mcmc diagnostics all look good.
List model variable names
get_variables(m1b) [1] "(Intercept)" "length_z" "snowt_z" "snowt1_z" "day_z" "bdload_z" "elev_z" "first_c0.353"
[9] "male_c0.548" "donor70567" "donor72996" "accept_stat__" "stepsize__" "treedepth__" "n_leapfrog__" "divergent__"
[17] "energy__" Posterior predictive check
pp_check(m1b, plotfun = "stat") +
xlab("Frog survival rate")
- Posterior predictive distribution implied by regression model closely matches original data (based on mean frog survival).
Calculate loo
loo_m1b <- loo(m1b, cores = 4, save_psis = TRUE)
print(loo_m1b)
Computed from 20000 by 767 log-likelihood matrix------
Monte Carlo SE of elpd_loo is 0.0.
All Pareto k estimates are good (k < 0.5).
See help('pareto-k-diagnostic') for details.Posterior analysis
tidy(m1b, conf.int = TRUE, conf.level = 0.90)
mcmc_areas(m1b, area_method = "equal height", prob = 0.90) +
geom_vline(xintercept = 0, color = "red", linetype = "dashed", lwd = 0.5) +
ggtitle("Model with standardized predictors only",
"Posterior distributions with medians & 90% uncertainty intervals")
mean(bayes_R2(m1b))[1] 0.2728229- As expected, using standardized predictors allows easy comparison across predictors. Important predictors are unchanged.
- Based on 90% uncertainty intervals, important predictors are snow_t1, elevation, first, donor70567, and donor72996.
Compare models
loo_compare(loo_m1a, loo_m1b) elpd_diff se_diff
m1a 0.0 0.0
m1b 0.0 0.0 - As expected, models with non-standardized and standardized predictor variables have identical predictive accuracy.
Possible grouping structures to add to model
The dataset has three groups: site_id, translocation_id, and donor site. Donor site is not suitable as a grouping variable because there are only 3 levels (i.e., site ids), not enough to make this useful. Site_id and translocation_id contain 12 and 24 levels, respectively, making them suitable as grouping variables. Including both site_id and translocation_id as nested grouping variables (i.e., translocations nested within site ids) also seems justified. Having more levels within a group would seem to increase the power to detect effects of predictors. With fewer levels/less variability, effect may be accounted for primarily by grouping variable itself instead of predictor. With only 1-4 translocations per site and only 12 site ids total, it is possible that model with site_id as sole grouping variable and model with site_id and translocation_id as nested grouping variables will attribute more of the variation in frog survival to the grouping variables and less to the predictors. A priori, that would suggest that the model with translocation_id as the grouping variable might be the most useful model structure (assuming that all models have similar posterior predictive accuracy). Lacking any strong rationale for one model structure over another, built 3 models and compared their posterior predictive accuracy using loo.
Model using standardized predictors and site_id grouping variable
Model specification
m1c <- stan_glmer(
surv ~ length_z + snowt_z + snowt1_z + day_z + bdload_z + elev_z + first_c + male_c + donor + (1 | site_id),
data = d1,
family = binomial,
chains = 4,
iter = 5000*2,
cores = 4,
seed = 84735)starting worker pid=11308 on localhost:11976 at 13:35:38.500
starting worker pid=11337 on localhost:11976 at 13:35:38.651
starting worker pid=11372 on localhost:11976 at 13:35:38.790
starting worker pid=11403 on localhost:11976 at 13:35:38.937
SAMPLING FOR MODEL 'bernoulli' NOW (CHAIN 1).
Chain 1: Rejecting initial value:
Chain 1: Log probability evaluates to log(0), i.e. negative infinity.
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Chain 1:
Chain 1: Gradient evaluation took 0.000169 seconds
Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 1.69 seconds.
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Chain 1: Priors & chain diagnostics
prior_summary(m1c)Priors for model 'm1c'
------
Intercept (after predictors centered)
~ normal(location = 0, scale = 2.5)
Coefficients
Specified prior:
~ normal(location = [0,0,0,...], scale = [2.5,2.5,2.5,...])
Adjusted prior:
~ normal(location = [0,0,0,...], scale = [5.00,5.00,5.00,...])
Covariance
~ decov(reg. = 1, conc. = 1, shape = 1, scale = 1)
------
See help('prior_summary.stanreg') for more detailsb1 <- c("(Intercept)", "length_z", "snowt_z", "snowt1_z", "day_z", "bdload_z", "elev_z", "first_c0.353", "male_c0.548", "donor70567", "donor72996")
mcmc_trace(m1c, size = 0.1, pars = b1)
mcmc_dens_overlay(m1c, pars = b1)
summary(m1c)
Model Info:
function: stan_glmer
family: binomial [logit]
formula: surv ~ length_z + snowt_z + snowt1_z + day_z + bdload_z + elev_z +
first_c + male_c + donor + (1 | site_id)
algorithm: sampling
sample: 20000 (posterior sample size)
priors: see help('prior_summary')
observations: 767
groups: site_id (12)
Estimates:
mean sd 10% 50% 90%
(Intercept) 0.0 1.2 -1.5 0.0 1.3
length_z 0.0 0.2 -0.3 0.0 0.3
snowt_z 0.0 0.4 -0.5 0.0 0.5
snowt1_z 0.0 0.5 -0.7 -0.1 0.7
day_z 0.2 0.4 -0.4 0.2 0.7
bdload_z -0.2 0.2 -0.5 -0.2 0.1
elev_z 1.7 1.0 0.5 1.7 3.0
first_c0.353 0.3 0.2 0.0 0.3 0.7
male_c0.548 0.4 0.2 0.1 0.3 0.6
donor70567 -0.8 1.7 -2.8 -0.8 1.3
donor72996 -1.9 1.3 -3.4 -1.9 -0.3
b[(Intercept) site_id:70134] -0.4 0.6 -1.2 -0.4 0.3
b[(Intercept) site_id:70370] -1.7 1.0 -2.9 -1.6 -0.5
b[(Intercept) site_id:70413] 0.0 1.3 -1.6 0.0 1.5
b[(Intercept) site_id:70414] -1.2 1.2 -2.7 -1.0 0.2
b[(Intercept) site_id:70449] 1.5 0.6 0.8 1.5 2.3
b[(Intercept) site_id:70505] 0.2 0.6 -0.5 0.2 1.0
b[(Intercept) site_id:70550] 0.6 0.6 -0.2 0.5 1.4
b[(Intercept) site_id:70556] -0.9 1.0 -2.2 -0.9 0.4
b[(Intercept) site_id:70619] -0.6 0.7 -1.5 -0.5 0.3
b[(Intercept) site_id:70628] 1.1 0.8 0.2 1.1 2.1
b[(Intercept) site_id:70641] 0.0 0.8 -0.9 0.0 1.0
b[(Intercept) site_id:74976] 1.0 1.0 -0.2 0.9 2.3
Sigma[site_id:(Intercept),(Intercept)] 1.9 1.4 0.6 1.5 3.5
Fit Diagnostics:
mean sd 10% 50% 90%
mean_PPD 0.3 0.0 0.3 0.3 0.3
The mean_ppd is the sample average posterior predictive distribution of the outcome variable (for details see help('summary.stanreg')).
MCMC diagnostics
mcse Rhat n_eff
(Intercept) 0.0 1.0 9604
length_z 0.0 1.0 29640
snowt_z 0.0 1.0 12471
snowt1_z 0.0 1.0 9663
day_z 0.0 1.0 9895
bdload_z 0.0 1.0 29229
elev_z 0.0 1.0 8370
first_c0.353 0.0 1.0 25887
male_c0.548 0.0 1.0 30649
donor70567 0.0 1.0 12665
donor72996 0.0 1.0 9205
b[(Intercept) site_id:70134] 0.0 1.0 12243
b[(Intercept) site_id:70370] 0.0 1.0 9052
b[(Intercept) site_id:70413] 0.0 1.0 16783
b[(Intercept) site_id:70414] 0.0 1.0 13633
b[(Intercept) site_id:70449] 0.0 1.0 9662
b[(Intercept) site_id:70505] 0.0 1.0 10089
b[(Intercept) site_id:70550] 0.0 1.0 9125
b[(Intercept) site_id:70556] 0.0 1.0 13480
b[(Intercept) site_id:70619] 0.0 1.0 9114
b[(Intercept) site_id:70628] 0.0 1.0 9365
b[(Intercept) site_id:70641] 0.0 1.0 8782
b[(Intercept) site_id:74976] 0.0 1.0 10720
Sigma[site_id:(Intercept),(Intercept)] 0.0 1.0 7013
mean_PPD 0.0 1.0 18973
log-posterior 0.1 1.0 5668
For each parameter, mcse is Monte Carlo standard error, n_eff is a crude measure of effective sample size, and Rhat is the potential scale reduction factor on split chains (at convergence Rhat=1).Posterior predictive check
pp_check(m1c, plotfun = "stat") +
xlab("Frog survival rate")
Calculate loo
loo_m1c <- loo(m1c, cores = 4, save_psis = TRUE)
print(loo_m1c)
Computed from 20000 by 767 log-likelihood matrix------
Monte Carlo SE of elpd_loo is 0.0.
All Pareto k estimates are good (k < 0.5).
See help('pareto-k-diagnostic') for details.Posterior analysis
tidy(m1c, conf.int = TRUE, conf.level = 0.90)
mcmc_areas(m1c, area_method = "equal height", prob = 0.90, pars = b1) +
geom_vline(xintercept = 0, color = "red", linetype = "dashed", lwd = 0.5) +
ggtitle("Model with standardized predictors & site_id as grouping variable",
"Posterior distributions with medians & 90% uncertainty intervals")
mean(bayes_R2(m1c))[1] 0.3169546# Frog survival by site_id
mcmc_areas_ridges(m1c, regex_pars = "site_id") +
geom_vline(xintercept = 0, color = "red", linetype = "dashed", lwd = 0.5) +
ggtitle("Frog survival for each recipient site_id")
- Based on 90% uncertainty intervals, important predictors reduced to elevation and male.
Model using standardized predictors and translocation_id grouping variable)
Model specification
m1d <- stan_glmer(
surv ~ length_z + snowt_z + snowt1_z + day_z + bdload_z + elev_z + first_c + male_c + donor + (1 | translocation_id),
data = d1,
family = binomial,
chains = 4,
iter = 5000*2,
cores = 4,
seed = 84735)starting worker pid=11797 on localhost:11976 at 13:37:04.696
starting worker pid=11826 on localhost:11976 at 13:37:04.820
starting worker pid=11855 on localhost:11976 at 13:37:04.938
starting worker pid=11884 on localhost:11976 at 13:37:05.060
SAMPLING FOR MODEL 'bernoulli' NOW (CHAIN 1).
Chain 1: Rejecting initial value:
Chain 1: Log probability evaluates to log(0), i.e. negative infinity.
Chain 1: Stan can't start sampling from this initial value.
Chain 1:
Chain 1: Gradient evaluation took 0.000111 seconds
Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 1.11 seconds.
Chain 1: Adjust your expectations accordingly!
Chain 1:
Chain 1:
Chain 1: Iteration: 1 / 10000 [ 0%] (Warmup)
SAMPLING FOR MODEL 'bernoulli' NOW (CHAIN 2).
Chain 2:
Chain 2: Gradient evaluation took 0.000108 seconds
Chain 2: 1000 transitions using 10 leapfrog steps per transition would take 1.08 seconds.
Chain 2: Adjust your expectations accordingly!
Chain 2:
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SAMPLING FOR MODEL 'bernoulli' NOW (CHAIN 3).
Chain 3:
Chain 3: Gradient evaluation took 0.00012 seconds
Chain 3: 1000 transitions using 10 leapfrog steps per transition would take 1.2 seconds.
Chain 3: Adjust your expectations accordingly!
Chain 3:
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SAMPLING FOR MODEL 'bernoulli' NOW (CHAIN 4).
Chain 4:
Chain 4: Gradient evaluation took 0.000115 seconds
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Chain 4: Priors & chain diagnostics
prior_summary(m1d)Priors for model 'm1d'
------
Intercept (after predictors centered)
~ normal(location = 0, scale = 2.5)
Coefficients
Specified prior:
~ normal(location = [0,0,0,...], scale = [2.5,2.5,2.5,...])
Adjusted prior:
~ normal(location = [0,0,0,...], scale = [5.00,5.00,5.00,...])
Covariance
~ decov(reg. = 1, conc. = 1, shape = 1, scale = 1)
------
See help('prior_summary.stanreg') for more detailsb1 <- c("(Intercept)", "length_z", "snowt_z", "snowt1_z", "day_z", "bdload_z", "elev_z", "first_c0.353", "male_c0.548", "donor70567", "donor72996")
mcmc_trace(m1d, size = 0.1, pars = b1)
mcmc_dens_overlay(m1d, pars = b1)
summary(m1d)
Model Info:
function: stan_glmer
family: binomial [logit]
formula: surv ~ length_z + snowt_z + snowt1_z + day_z + bdload_z + elev_z +
first_c + male_c + donor + (1 | translocation_id)
algorithm: sampling
sample: 20000 (posterior sample size)
priors: see help('prior_summary')
observations: 767
groups: translocation_id (24)
Estimates:
mean sd 10% 50% 90%
(Intercept) 0.3 0.7 -0.6 0.3 1.2
length_z 0.0 0.2 -0.3 0.0 0.3
snowt_z 0.0 0.6 -0.7 0.0 0.7
snowt1_z -1.2 0.5 -1.9 -1.2 -0.5
day_z 0.1 0.5 -0.6 0.1 0.8
bdload_z -0.2 0.2 -0.5 -0.2 0.0
elev_z 1.6 0.6 0.8 1.6 2.4
first_c0.353 0.5 0.5 -0.1 0.5 1.1
male_c0.548 0.3 0.2 0.1 0.3 0.6
donor70567 -1.3 0.9 -2.4 -1.3 -0.2
donor72996 -2.2 0.8 -3.2 -2.2 -1.3
b[(Intercept) translocation_id:70134_1] -0.6 0.5 -1.3 -0.6 0.1
b[(Intercept) translocation_id:70370_1] -0.6 0.8 -1.6 -0.5 0.4
b[(Intercept) translocation_id:70370_2] 0.0 0.7 -0.9 0.0 0.9
b[(Intercept) translocation_id:70413_1] -0.1 0.7 -1.0 -0.1 0.8
b[(Intercept) translocation_id:70413_2] 0.4 0.7 -0.5 0.4 1.3
b[(Intercept) translocation_id:70413_3] -0.3 0.8 -1.3 -0.3 0.6
b[(Intercept) translocation_id:70414_1] -0.9 0.8 -2.0 -0.9 0.1
b[(Intercept) translocation_id:70449_1] 0.1 0.6 -0.7 0.1 0.9
b[(Intercept) translocation_id:70449_2] 1.6 0.6 0.9 1.6 2.4
b[(Intercept) translocation_id:70505_1] 0.1 0.5 -0.6 0.1 0.8
b[(Intercept) translocation_id:70505_2] 0.1 0.6 -0.7 0.1 0.9
b[(Intercept) translocation_id:70505_3] 0.2 0.7 -0.7 0.2 1.0
b[(Intercept) translocation_id:70505_4] -0.1 0.6 -0.9 -0.1 0.7
b[(Intercept) translocation_id:70550_1] 0.5 0.5 -0.2 0.5 1.2
b[(Intercept) translocation_id:70550_2] -0.1 0.6 -0.9 -0.1 0.6
b[(Intercept) translocation_id:70556_1] -0.5 0.7 -1.4 -0.5 0.4
b[(Intercept) translocation_id:70556_2] -0.8 0.7 -1.8 -0.8 0.0
b[(Intercept) translocation_id:70619_1] -0.5 0.6 -1.2 -0.5 0.2
b[(Intercept) translocation_id:70628_1] 0.8 0.6 0.0 0.8 1.6
b[(Intercept) translocation_id:70641_1] 0.2 0.7 -0.7 0.2 1.1
b[(Intercept) translocation_id:70641_2] -0.3 0.7 -1.1 -0.2 0.6
b[(Intercept) translocation_id:70641_3] -0.7 0.7 -1.7 -0.7 0.2
b[(Intercept) translocation_id:74976_1] 1.4 0.8 0.4 1.3 2.4
b[(Intercept) translocation_id:74976_2] 0.0 0.7 -0.8 0.0 0.9
Sigma[translocation_id:(Intercept),(Intercept)] 0.9 0.5 0.4 0.8 1.6
Fit Diagnostics:
mean sd 10% 50% 90%
mean_PPD 0.3 0.0 0.3 0.3 0.3
The mean_ppd is the sample average posterior predictive distribution of the outcome variable (for details see help('summary.stanreg')).
MCMC diagnostics
mcse Rhat n_eff
(Intercept) 0.0 1.0 10818
length_z 0.0 1.0 25683
snowt_z 0.0 1.0 10328
snowt1_z 0.0 1.0 12115
day_z 0.0 1.0 10672
bdload_z 0.0 1.0 28564
elev_z 0.0 1.0 11059
first_c0.353 0.0 1.0 11342
male_c0.548 0.0 1.0 26534
donor70567 0.0 1.0 10846
donor72996 0.0 1.0 10753
b[(Intercept) translocation_id:70134_1] 0.0 1.0 14706
b[(Intercept) translocation_id:70370_1] 0.0 1.0 15110
b[(Intercept) translocation_id:70370_2] 0.0 1.0 17141
b[(Intercept) translocation_id:70413_1] 0.0 1.0 13738
b[(Intercept) translocation_id:70413_2] 0.0 1.0 14123
b[(Intercept) translocation_id:70413_3] 0.0 1.0 16112
b[(Intercept) translocation_id:70414_1] 0.0 1.0 17969
b[(Intercept) translocation_id:70449_1] 0.0 1.0 12532
b[(Intercept) translocation_id:70449_2] 0.0 1.0 13621
b[(Intercept) translocation_id:70505_1] 0.0 1.0 14823
b[(Intercept) translocation_id:70505_2] 0.0 1.0 18847
b[(Intercept) translocation_id:70505_3] 0.0 1.0 20672
b[(Intercept) translocation_id:70505_4] 0.0 1.0 17878
b[(Intercept) translocation_id:70550_1] 0.0 1.0 11544
b[(Intercept) translocation_id:70550_2] 0.0 1.0 15234
b[(Intercept) translocation_id:70556_1] 0.0 1.0 14110
b[(Intercept) translocation_id:70556_2] 0.0 1.0 14298
b[(Intercept) translocation_id:70619_1] 0.0 1.0 12428
b[(Intercept) translocation_id:70628_1] 0.0 1.0 11195
b[(Intercept) translocation_id:70641_1] 0.0 1.0 11049
b[(Intercept) translocation_id:70641_2] 0.0 1.0 18120
b[(Intercept) translocation_id:70641_3] 0.0 1.0 16196
b[(Intercept) translocation_id:74976_1] 0.0 1.0 14344
b[(Intercept) translocation_id:74976_2] 0.0 1.0 14465
Sigma[translocation_id:(Intercept),(Intercept)] 0.0 1.0 8036
mean_PPD 0.0 1.0 19612
log-posterior 0.1 1.0 5727
For each parameter, mcse is Monte Carlo standard error, n_eff is a crude measure of effective sample size, and Rhat is the potential scale reduction factor on split chains (at convergence Rhat=1).Posterior predictive check
pp_check(m1d, plotfun = "stat") +
xlab("Frog survival rate")
Calculate loo
loo_m1d <- loo(m1d, cores = 4, save_psis = TRUE)
print(loo_m1d)
Computed from 20000 by 767 log-likelihood matrix------
Monte Carlo SE of elpd_loo is 0.0.
All Pareto k estimates are good (k < 0.5).
See help('pareto-k-diagnostic') for details.Posterior analysis
tidy(m1d, conf.int = TRUE, conf.level = 0.90)
mcmc_areas(m1d, area_method = "equal height", prob = 0.90, pars = b1) +
geom_vline(xintercept = 0, color = "red", linetype = "dashed", lwd = 0.5) +
ggtitle("Model with standardized predictors & translocation_id as grouping variable",
"Posterior distributions with medians & 90% uncertainty intervals")
mean(bayes_R2(m1d))[1] 0.322301# Frog survival by translocation_id
mcmc_areas_ridges(m1d, regex_pars = "translocation_id") +
geom_vline(xintercept = 0, color = "red", linetype = "dashed", lwd = 0.5) +
ggtitle("Frog survival for each translocation_id")
- Based on loo, model with translocation_id as grouping variable provides slightly better fit than model with site_id as grouping variable.
- Based on 90% uncertainty intervals, important predictors are similar to those in model without grouping variable: snowt1_z, elev_z, male_c0.548, donor72996.
Model using standardized predictors and 2 grouping variables: translocation_id nested within site_id)
Model specification
m1e <- stan_glmer(
surv ~ length_z + snowt_z + snowt1_z + day_z + bdload_z + elev_z + first_c + male_c + donor + (1 | translocation_id) + (1 | site_id),
data = d1,
family = binomial,
chains = 4,
iter = 5000*2,
cores = 4,
seed = 84735)starting worker pid=12308 on localhost:11976 at 13:38:14.420
starting worker pid=12337 on localhost:11976 at 13:38:14.541
starting worker pid=12366 on localhost:11976 at 13:38:14.669
starting worker pid=12395 on localhost:11976 at 13:38:14.798
SAMPLING FOR MODEL 'bernoulli' NOW (CHAIN 1).
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Chain 4: Priors & chain diagnostics
prior_summary(m1e)Priors for model 'm1e'
------
Intercept (after predictors centered)
~ normal(location = 0, scale = 2.5)
Coefficients
Specified prior:
~ normal(location = [0,0,0,...], scale = [2.5,2.5,2.5,...])
Adjusted prior:
~ normal(location = [0,0,0,...], scale = [5.00,5.00,5.00,...])
Covariance
~ decov(reg. = 1, conc. = 1, shape = 1, scale = 1)
------
See help('prior_summary.stanreg') for more detailsb1 <- c("(Intercept)", "length_z", "snowt_z", "snowt1_z", "day_z", "bdload_z", "elev_z", "first_c0.353", "male_c0.548", "donor70567", "donor72996")
mcmc_trace(m1e, size = 0.1, pars = b1)
mcmc_dens_overlay(m1e, pars = b1)
summary(m1e)
Model Info:
function: stan_glmer
family: binomial [logit]
formula: surv ~ length_z + snowt_z + snowt1_z + day_z + bdload_z + elev_z +
first_c + male_c + donor + (1 | translocation_id) + (1 |
site_id)
algorithm: sampling
sample: 20000 (posterior sample size)
priors: see help('prior_summary')
observations: 767
groups: translocation_id (24), site_id (12)
Estimates:
mean sd 10% 50% 90%
(Intercept) 0.1 1.0 -1.2 0.1 1.3
length_z 0.0 0.2 -0.3 0.0 0.3
snowt_z -0.1 0.5 -0.7 -0.1 0.6
snowt1_z -0.5 0.7 -1.5 -0.6 0.4
day_z 0.2 0.5 -0.5 0.2 0.9
bdload_z -0.2 0.2 -0.5 -0.2 0.0
elev_z 1.7 0.9 0.6 1.7 2.9
first_c0.353 0.4 0.4 0.0 0.4 0.9
male_c0.548 0.3 0.2 0.1 0.3 0.6
donor70567 -1.0 1.5 -2.7 -1.1 0.8
donor72996 -2.0 1.1 -3.4 -2.1 -0.7
b[(Intercept) translocation_id:70134_1] -0.2 0.5 -0.9 -0.1 0.4
b[(Intercept) translocation_id:70370_1] -0.4 0.6 -1.1 -0.3 0.2
b[(Intercept) translocation_id:70370_2] 0.1 0.5 -0.5 0.1 0.7
b[(Intercept) translocation_id:70413_1] 0.0 0.5 -0.6 0.0 0.6
b[(Intercept) translocation_id:70413_2] 0.2 0.5 -0.4 0.1 0.9
b[(Intercept) translocation_id:70413_3] -0.2 0.6 -0.9 -0.2 0.4
b[(Intercept) translocation_id:70414_1] -0.3 0.7 -1.2 -0.2 0.3
b[(Intercept) translocation_id:70449_1] -0.1 0.5 -0.7 -0.1 0.5
b[(Intercept) translocation_id:70449_2] 0.7 0.7 0.0 0.6 1.7
b[(Intercept) translocation_id:70505_1] 0.0 0.5 -0.5 0.0 0.6
b[(Intercept) translocation_id:70505_2] 0.1 0.5 -0.5 0.1 0.7
b[(Intercept) translocation_id:70505_3] 0.1 0.5 -0.5 0.0 0.7
b[(Intercept) translocation_id:70505_4] -0.1 0.5 -0.7 -0.1 0.5
b[(Intercept) translocation_id:70550_1] 0.3 0.5 -0.3 0.2 0.9
b[(Intercept) translocation_id:70550_2] -0.1 0.5 -0.7 -0.1 0.4
b[(Intercept) translocation_id:70556_1] -0.2 0.6 -0.9 -0.1 0.4
b[(Intercept) translocation_id:70556_2] -0.2 0.6 -1.0 -0.1 0.4
b[(Intercept) translocation_id:70619_1] -0.2 0.5 -0.9 -0.1 0.4
b[(Intercept) translocation_id:70628_1] 0.3 0.6 -0.3 0.2 1.1
b[(Intercept) translocation_id:70641_1] 0.2 0.5 -0.4 0.1 0.9
b[(Intercept) translocation_id:70641_2] -0.1 0.5 -0.7 0.0 0.5
b[(Intercept) translocation_id:70641_3] -0.3 0.6 -1.1 -0.2 0.2
b[(Intercept) translocation_id:74976_1] 0.5 0.7 -0.1 0.4 1.5
b[(Intercept) translocation_id:74976_2] -0.1 0.5 -0.7 -0.1 0.5
b[(Intercept) site_id:70134] -0.3 0.6 -1.1 -0.3 0.4
b[(Intercept) site_id:70370] -0.9 1.0 -2.3 -0.7 0.2
b[(Intercept) site_id:70413] 0.0 1.1 -1.3 0.0 1.2
b[(Intercept) site_id:70414] -0.7 1.1 -2.1 -0.5 0.3
b[(Intercept) site_id:70449] 1.0 0.8 0.0 1.0 2.0
b[(Intercept) site_id:70505] 0.2 0.6 -0.5 0.1 0.9
b[(Intercept) site_id:70550] 0.3 0.6 -0.4 0.2 1.1
b[(Intercept) site_id:70556] -0.7 0.9 -1.8 -0.5 0.3
b[(Intercept) site_id:70619] -0.4 0.7 -1.3 -0.3 0.4
b[(Intercept) site_id:70628] 0.6 0.8 -0.2 0.5 1.7
b[(Intercept) site_id:70641] -0.1 0.7 -1.0 -0.1 0.7
b[(Intercept) site_id:74976] 0.7 0.9 -0.2 0.6 1.9
Sigma[translocation_id:(Intercept),(Intercept)] 0.4 0.4 0.0 0.3 0.9
Sigma[site_id:(Intercept),(Intercept)] 1.2 1.3 0.1 0.8 2.6
Fit Diagnostics:
mean sd 10% 50% 90%
mean_PPD 0.3 0.0 0.3 0.3 0.3
The mean_ppd is the sample average posterior predictive distribution of the outcome variable (for details see help('summary.stanreg')).
MCMC diagnostics
mcse Rhat n_eff
(Intercept) 0.0 1.0 8760
length_z 0.0 1.0 26927
snowt_z 0.0 1.0 12055
snowt1_z 0.0 1.0 5542
day_z 0.0 1.0 10670
bdload_z 0.0 1.0 28589
elev_z 0.0 1.0 10334
first_c0.353 0.0 1.0 12870
male_c0.548 0.0 1.0 27066
donor70567 0.0 1.0 9522
donor72996 0.0 1.0 8564
b[(Intercept) translocation_id:70134_1] 0.0 1.0 10178
b[(Intercept) translocation_id:70370_1] 0.0 1.0 13784
b[(Intercept) translocation_id:70370_2] 0.0 1.0 21315
b[(Intercept) translocation_id:70413_1] 0.0 1.0 18159
b[(Intercept) translocation_id:70413_2] 0.0 1.0 16405
b[(Intercept) translocation_id:70413_3] 0.0 1.0 18489
b[(Intercept) translocation_id:70414_1] 0.0 1.0 8310
b[(Intercept) translocation_id:70449_1] 0.0 1.0 15651
b[(Intercept) translocation_id:70449_2] 0.0 1.0 4203
b[(Intercept) translocation_id:70505_1] 0.0 1.0 18690
b[(Intercept) translocation_id:70505_2] 0.0 1.0 23587
b[(Intercept) translocation_id:70505_3] 0.0 1.0 23491
b[(Intercept) translocation_id:70505_4] 0.0 1.0 22403
b[(Intercept) translocation_id:70550_1] 0.0 1.0 12418
b[(Intercept) translocation_id:70550_2] 0.0 1.0 19758
b[(Intercept) translocation_id:70556_1] 0.0 1.0 14624
b[(Intercept) translocation_id:70556_2] 0.0 1.0 7455
b[(Intercept) translocation_id:70619_1] 0.0 1.0 11194
b[(Intercept) translocation_id:70628_1] 0.0 1.0 8502
b[(Intercept) translocation_id:70641_1] 0.0 1.0 15452
b[(Intercept) translocation_id:70641_2] 0.0 1.0 20501
b[(Intercept) translocation_id:70641_3] 0.0 1.0 9364
b[(Intercept) translocation_id:74976_1] 0.0 1.0 5537
b[(Intercept) translocation_id:74976_2] 0.0 1.0 20547
b[(Intercept) site_id:70134] 0.0 1.0 12998
b[(Intercept) site_id:70370] 0.0 1.0 5422
b[(Intercept) site_id:70413] 0.0 1.0 14150
b[(Intercept) site_id:70414] 0.0 1.0 9926
b[(Intercept) site_id:70449] 0.0 1.0 4422
b[(Intercept) site_id:70505] 0.0 1.0 10980
b[(Intercept) site_id:70550] 0.0 1.0 9459
b[(Intercept) site_id:70556] 0.0 1.0 10122
b[(Intercept) site_id:70619] 0.0 1.0 9424
b[(Intercept) site_id:70628] 0.0 1.0 6708
b[(Intercept) site_id:70641] 0.0 1.0 9372
b[(Intercept) site_id:74976] 0.0 1.0 7086
Sigma[translocation_id:(Intercept),(Intercept)] 0.0 1.0 4082
Sigma[site_id:(Intercept),(Intercept)] 0.0 1.0 4867
mean_PPD 0.0 1.0 19556
log-posterior 0.1 1.0 5917
For each parameter, mcse is Monte Carlo standard error, n_eff is a crude measure of effective sample size, and Rhat is the potential scale reduction factor on split chains (at convergence Rhat=1).Posterior predictive check
pp_check(m1e, plotfun = "stat") +
xlab("Frog survival rate")
Calculate loo
loo_m1e <- loo(m1e, cores = 4, save_psis = TRUE)
print(loo_m1e)
Computed from 20000 by 767 log-likelihood matrix------
Monte Carlo SE of elpd_loo is 0.0.
All Pareto k estimates are good (k < 0.5).
See help('pareto-k-diagnostic') for details.Posterior analysis
tidy(m1e, conf.int = TRUE, conf.level = 0.90)
mcmc_areas(m1e, area_method = "equal height", prob = 0.90, pars = b1) +
geom_vline(xintercept = 0, color = "red", linetype = "dashed", lwd = 0.5) +
ggtitle("Model with standardized predictors & translocation_id as grouping variable",
"Posterior distributions with medians & 90% uncertainty intervals")
mean(bayes_R2(m1e))[1] 0.3232688# Frog survival by translocation_id
mcmc_areas_ridges(m1e, regex_pars = ("translocation_id")) +
geom_vline(xintercept = 0, color = "red", linetype = "dashed", lwd = 0.5) +
ggtitle("Frog survival for each translocation_id")
# Frog survival by site_id
mcmc_areas_ridges(m1e, regex_pars = ("site_id")) +
geom_vline(xintercept = 0, color = "red", linetype = "dashed", lwd = 0.5) +
ggtitle("Frog survival for each translocation_id")
Compare 4 models with different grouping structures
loo_compare(loo_m1b, loo_m1c, loo_m1d, loo_m1e) elpd_diff se_diff
m1c 0.0 0.0
m1e -0.5 1.9
m1d -1.1 3.1
m1b -18.2 6.3 - Based on expected log-predictive densities (ELPD), 3 models with grouping variables all have substantially higher posterior predictive accuracy than model without any grouping variables (elpd_diff of m1b is more than 2 SE lower than elpd_diff of m1c: elpd_diff = -18.2 \(\pm\) 12.6, 2 SE interval does not overlap 0: -5.6 – -30.8).
- 3 models with 1 or more grouping variables are equivalent and have
similar posterior predictive accuracy (elpd_diff of m1e and m1d are
within 2 SE of elpd_diff of m1c).
- m1d seems to provide more insight into the effect of the predictors, so chose this model for additional analysis.
Additional analysis of model m1d
Posterior classification accuracy
# Posterior predictive models for each frog in dataset
set.seed(84735)
survival_predict_1 <- posterior_predict(m1d, newdata = d1)
# Classify frog survival based on mean predicted survival probability
survival_predict_2 <- d1 %>%
mutate(survival_prob = colMeans(survival_predict_1),
survival_class = as.numeric(survival_prob >= 0.5)) %>%
select(site_id, translocation_id, year, snowt1_z, elev_z, male_c, donor, surv, survival_prob, survival_class)
# Generate confusion matrix and calculate accuracy
survival_predict_2 %>%
tabyl(surv, survival_class) %>%
adorn_totals(c("row", "col")) surv 0 1 Total
0 471 62 533
1 96 138 234
Total 567 200 767(471 + 138)/767 # overall_accuracy [1] 0.7940026471/533 # true negative rate[1] 0.8836773138/234 # true positive rate[1] 0.5897436# Generate confusion matrix using function
set.seed(84735)
classification_summary(m1d, data = d1, cutoff = 0.5)$confusion_matrix
y 0 1
0 471 62
1 96 138
$accuracy_ratesNA- Sensitivity or “true positive rate” measures the proportion of Y = 1
observations that are accurately classified.
- Specificity or “true negative rate” measures the proportion of Y = 0 observations that are accurately classified (i.e., proportion of frogs correctly identified as not surviving).
Create reduced model containing only important predictors for subsequent analyses
- Reduced model is helpful because it reduces the number of values used to calculate predictive models.
m1d_reduce <- stan_glmer(
surv ~ snowt1_z + elev_z + male_c + donor + (1 | translocation_id),
data = d1,
family = binomial,
chains = 4,
iter = 5000*2,
cores = 4,
seed = 84735)starting worker pid=12738 on localhost:11976 at 13:40:15.905
starting worker pid=12767 on localhost:11976 at 13:40:16.051
starting worker pid=12796 on localhost:11976 at 13:40:16.183
starting worker pid=12825 on localhost:11976 at 13:40:16.313
SAMPLING FOR MODEL 'bernoulli' NOW (CHAIN 1).
Chain 1:
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Chain 1: Adjust your expectations accordingly!
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SAMPLING FOR MODEL 'bernoulli' NOW (CHAIN 2).
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Chain 2: Gradient evaluation took 0.000101 seconds
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SAMPLING FOR MODEL 'bernoulli' NOW (CHAIN 3).
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SAMPLING FOR MODEL 'bernoulli' NOW (CHAIN 4).
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Chain 4: Confirm fit of reduced model is comparable to full model
loo_m1d_reduce <- loo(m1d, cores = 4, save_psis = TRUE)
print(loo_m1d)
Computed from 20000 by 767 log-likelihood matrix------
Monte Carlo SE of elpd_loo is 0.0.
All Pareto k estimates are good (k < 0.5).
See help('pareto-k-diagnostic') for details.loo_compare(loo_m1d, loo_m1d_reduce) elpd_diff se_diff
m1d 0.0 0.0
m1d 0.0 0.0 Simulate and plot posterior predictive models & observations
m1d_group_means <- d1 %>%
group_by(translocation_id) %>%
summarize(count = n(), psurv = mean(surv), elev_z = mean(elev_z), snowt1_z = mean(snowt1_z)) %>%
mutate(
male_c = as.factor(0.548),
donor = case_when(
str_detect(translocation_id, "74976") | str_detect(translocation_id, "70556") ~ "70459",
str_detect(translocation_id, "70413") ~ "70567",
TRUE ~ "72996")) %>%
arrange(psurv) # to arrange plot values/labels by psurv
predictions_m1d <- posterior_predict(m1d_reduce, newdata = m1d_group_means)
ppc_intervals(
y = m1d_group_means$psurv,
yrep = predictions_m1d,
prob = 0.5,
prob_outer = 0.8) +
ggplot2::scale_x_continuous(
labels = m1d_group_means$translocation_id,
breaks = 1:nrow(m1d_group_means)) +
xaxis_text(angle = 90, vjust = 0.5) +
xlab("Translocation_id")
- Uncertainty intervals appear incorrect for binomial model, but not
sure how to adjust.
- Based on mean values, model does an adequate job of predicting
survival based on group-level and individual-level
characteristics.
- What explains the low predicted survival at 70370 (both
translocations), and high versus low predicted survival at 70550 for 1st
and 2nd translocations, respectively? Both sites seem to contain high
quality habitat (i.e., high elevation).
- The predicted survival is based on assumption that only males are included. Run another set of models for females?
Plot posterior plausible relationships between each predictor and the probability of frog survival (conditional effects)
- As described in https://rdrr.io/cran/brms/man/conditional_effects.brmsfit.html,
“When creating conditional_effects for a particular predictor (or
interaction of two predictors), one has to choose the values of all
other predictors to condition on. By default, the mean is used for
continuous variables and the reference category is used for factors, but
you may change these values via argument conditions. This also has an
implication for the points argument: In the created plots, only those
points will be shown that correspond to the factor levels actually used
in the conditioning, in order not to create the false impression of bad
model fit, where it is just due to conditioning on certain factor
levels.”
- Using rstanarm, I was unable to do this as I expected to: Start with
data = d1 and using “newdata =” to specify that predictions should be
based on data in d1_elevz (for example), in which non-plotted continuous
predictors are set to the mean value. Not sure if my alternative
approach is correct.
- Approach used: Instead of starting the routine with d1, I started with the new dataset in which all variables except the one of interest were set to mean values (continuous variables) and the reference level (categorical variables). This produced plots/results that were very similar to those produced in brms using the conditional_effects function.
Effect of elevation
d1_elevz <- d1 %>%
mutate(snowt1_z = mean(snowt1_z),
male_c = recode(male_c, "-0.452" = "0.548"),
donor = recode(donor, "70567" = "70459", "72996" = "70459"))
d1_elevz %>% distinct(snowt1_z, male_c, donor) # ensure variables were set to correct values
d1_elevz %>%
select(site_id, translocation_id, snowt1_z, elev_z, male_c, donor, surv) %>%
add_epred_draws(m1d_reduce, ndraws = 100, re_formula = NA) %>%
ggplot(aes(x = elev_z, y = surv)) +
geom_line(aes(y = .epred, group = .draw), color = "blue", size = 0.2, alpha = 0.5) +
# geom_point(aes(y = .epred)) +
labs(x = "Elevation of recipient site (standardized)", y = "Probability of 1-year frog survival")
# d1 %>% # Test of add_predicted_draws used for similar purpose - unsuccessful
# select(site_id, translocation_id, snowt1_z, elev_z, male_c, donor, surv) %>%
# add_predicted_draws(m1d_reduce, ndraws = 100, re_formula = NA, newdata = d1_elevz) %>% View()
# ggplot(aes(x = elev_z, y = surv)) +
# geom_line(aes(y = .epred, group = .draw), color = "blue", size = 0.2, alpha = 0.5) +
# geom_point(aes(y = .epred)) +
# labs(x = "Elevation of recipient site (standardized)", y = "Probability of 1-year frog survival")Effect of winter severity in year t1
d1_snowt1z <- d1 %>%
mutate(elev_z = mean(elev_z),
male_c = recode(male_c, "-0.452" = "0.548"),
donor = recode(donor, "70567" = "70459", "72996" = "70459"))
d1_snowt1z %>% distinct(elev_z, male_c, donor) # ensure variables were set to correct values
d1_snowt1z %>%
select(site_id, translocation_id, snowt1_z, elev_z, male_c, donor, surv) %>%
add_epred_draws(m1d_reduce, ndraws = 100, re_formula = NA) %>%
ggplot(aes(x = snowt1_z, y = surv)) +
geom_line(aes(y = .epred, group = .draw), color = "blue", size = 0.2, alpha = 0.5) +
# geom_point(aes(y = .epred)) +
labs(x = "Winter severity in year following translocation (standardized)", y = "Probability of 1-year frog survival") 
Effect of sex
d1_malec <- d1 %>%
mutate(elev_z = mean(elev_z),
snowt1_z = mean(snowt1_z),
donor = recode(donor, "70567" = "70459", "72996" = "70459"))
d1_malec %>% distinct(elev_z, snowt1_z, donor) # ensure variables were set to correct values
d1_malec %>%
select(site_id, translocation_id, snowt1_z, elev_z, male_c, donor, surv) %>%
add_epred_draws(m1d_reduce, ndraws = 100, re_formula = NA) %>%
ggplot(aes(x = male_c, y = surv)) +
geom_boxplot(aes(y = .epred, x = male_c)) +
# geom_point(aes(y = .epred)) +
labs(x = "Sex (centered)", y = "Probability of 1-year frog survival")
Effect of donor
d1_donor <- d1 %>%
mutate(elev_z = mean(elev_z),
snowt1_z = mean(snowt1_z),
male_c = recode(male_c, "-0.452" = "0.548"))
d1_donor %>% distinct(elev_z, snowt1_z, male_c) # ensure variables were set to correct values
d1_donor %>%
select(site_id, translocation_id, snowt1_z, elev_z, male_c, donor, surv) %>%
add_epred_draws(m1d_reduce, ndraws = 100, re_formula = NA) %>%
ggplot(aes(x = donor, y = surv)) +
geom_boxplot(aes(y = .epred, x = donor)) +
# geom_point(aes(y = .epred)) +
labs(x = "Donor population", y = "Probability of 1-year frog survival")
Joint effects of elevation and sex
d1_elevz_malec <- d1 %>%
mutate(snowt1_z = mean(snowt1_z),
donor = recode(donor, "70567" = "70459", "72996" = "70459"))
d1_elevz_malec %>% distinct(snowt1_z, donor) # ensure variables were set to correct values
pal <- c("#0d0887", "#f89540")
d1_elevz_malec %>%
select(site_id, translocation_id, snowt1_z, elev_z, male_c, donor, surv) %>%
add_epred_draws(m1d_reduce, ndraws = 100, re_formula = NA) %>%
ggplot(aes(x = elev_z, y = surv, color = male_c)) +
geom_line(aes(y = .epred, group = paste(male_c, .draw)), size = 0.4, alpha = 0.5) +
# geom_point(aes(y = .epred)) +
labs(x = "Elevation of recipient site (standardized)", y = "Probability of 1-year frog survival") +
scale_color_manual(values = pal) +
theme_classic()
For comparison, fit model using brms and conditional_effects function to plot variable-specific effects
Fit model
m1d_reduce_brms <- brm(
surv ~ snowt1_z + elev_z + male_c + donor + (1 | translocation_id),
data = d1,
family = bernoulli(),
chains = 4,
iter = 5000*2,
cores = 4,
seed = 84735)Compiling Stan program...
Start samplingstarting worker pid=13412 on localhost:11976 at 13:41:34.174
starting worker pid=13441 on localhost:11976 at 13:41:34.302
starting worker pid=13470 on localhost:11976 at 13:41:34.439
starting worker pid=13499 on localhost:11976 at 13:41:34.568
SAMPLING FOR MODEL 'dbb8efe82731517ccb01114eea2991f6' NOW (CHAIN 1).
Chain 1:
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SAMPLING FOR MODEL 'dbb8efe82731517ccb01114eea2991f6' NOW (CHAIN 2).
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plot(conditional_effects(m1d_reduce_brms, spaghetti = TRUE, mean = TRUE, ndraws = 100), ask = FALSE)
TO DO
- Validate priors, tune as necessary. Add final priors to model
specification (in case the defaults change).
- Check approach to plotting conditional effects.
- Create analysis in which survival from first translocation to a site
is a predictor of survival from subsequent translocations. For this
analysis, will need to drop all sites at which only a single
translocation was conducted.
- Try un-standardizing predictor values in conditional-effects plots (and other plots?) to produce more intuitive visualizations.
---
title: "Predictors of frog survival following translocations - analyses"
author: "Roland Knapp"
output: html_notebook
---

The goal of the analyses in this notebook is to identify predictors of frog survival following translocation. The analyses are conducted in a Bayesian framework using the R package rstanarm.

## Load packages

```{r load-packages}
library(tidyverse)
library(brms)
library(rstanarm)
library(bayesplot)
library(tidybayes)
library(loo)
library(broom.mixed)
library(janitor)

source("classification_summary.R")
```

## Read in frog translocation dataset  
* Read in data, create variables for analysis, standardize continuous variables. 
* arm package used to rescale continuous variables (substract mean, divide by 2 SD), manual centering of binary variables.
```{r read-translocation-data}
d1 <- read_csv(here::here("data", "clean", "frog_translocation_final.csv")) %>% 
  drop_na(bd_load) %>%  # drop xx records from 70xxx where bd_load is NA
  mutate(
    first = if_else(order == 1, 1, 0),
    surv = if_else(survival < 0.5, 0, 1),
    male = if_else(sex == "m", 1, 0),
    elev_lo = if_else(elevation < 3020, 1, 0), # using interval defined by cut_interval, n = 2
    elev_z = arm::rescale(elevation),
    snowt_z = arm::rescale(snow_t),
    snowt1_z = arm::rescale(snow_t1),
    day_z = arm::rescale(day),
    length_z = arm::rescale(length),
    bdload_l = log10(bd_load + 1),
    bdload_z = arm::rescale(bdload_l),
    shore_c = shore - mean(shore),
    male_c = male - mean(male),
    elevlo_c = elev_lo - mean(elev_lo),
    first_c = first - mean(first),
    across(c(shore_c, male_c, elevlo_c, first_c), round, 3), 
    across(c(site_id, shore_c, first_c, donor, male_c, elevlo_c, year), as.factor),
    surv = as.integer(surv))

# Add a descriptive translocation_id
d1 <- d1 %>% 
  distinct(site_id, year) %>% 
  arrange(site_id, year) %>% 
  group_by(site_id) %>% 
  mutate(transno = seq_len(n())) %>% 
  ungroup(site_id) %>% 
  unite(translocation_id, c("site_id", "transno"), remove = FALSE) %>% 
  select(-transno) %>% 
  mutate(translocation_id = as.factor(translocation_id)) %>% 
  inner_join(d1, by = c("site_id", "year")) %>% 
  relocate(translocation_id, .after = site_id)
```

## Null values - check

```{r check-null}
d1 %>% summarize(across(everything(), ~ sum(is.na(.))))
```

## Model using non-standardized predictors
### Model specification
```{r}
m1a <- stan_glm(
  surv ~ length + snow_t + snow_t1 + day + bdload_l + elevation + first + male + donor,
  data = d1,
  family = binomial,
  chains = 4, 
  iter = 5000*2,
  cores = 4,
  seed = 84735)
```

### Priors & chain diagnostics
```{r}
prior_summary(m1a)

mcmc_trace(m1a, size = 0.1)
mcmc_dens_overlay(m1a)

summary(m1a)
```
- mcmc diagnostics all look good.

### List model variable names
```{r}
get_variables(m1a)
```

### Posterior predictive check
```{r}
pp_check(m1a, plotfun = "stat") +
    xlab("Frog survival rate")
```
- Posterior predictive distribution implied by regression model closely matches original data (based on mean frog survival).

### Calculate leave-one-out cross-validation (loo)
```{r}
loo_m1a <- loo(m1a, cores = 4, save_psis = TRUE)
print(loo_m1a)
```

### Posterior analysis
```{r}
tidy(m1a, conf.int = TRUE, conf.level = 0.90)

mcmc_intervals(m1a, point_est = "median", prob = 0.80, prob_outer = 0.9) +
  geom_vline(xintercept = 0, color = "red", linetype = "dashed", lwd = 0.5) +  # add 0-line to accentuate default line
  ggtitle("Model with non-standardized predictors only",
          "Posterior distributions with medians & 90% uncertainty intervals")

mcmc_areas(m1a, area_method = "equal height", prob = 0.90) +
  geom_vline(xintercept = 0, color = "red", linetype = "dashed", lwd = 0.5) +  
  ggtitle("Model with non-standardized predictors only",
          "Posterior distributions with medians & 90% uncertainty intervals")

mean(bayes_R2(m1a))
```
- Using non-standardized predictors makes plot hard to read due to vastly different scales of predictors.  
- Based on 90% uncertainty intervals, important predictors are snow_t1, elevation, first, donor70567, and donor72996.

## Model using standardized predictors
### Model specification
```{r}
m1b <- stan_glm(
  surv ~ length_z + snowt_z + snowt1_z + day_z + bdload_z + elev_z + first_c + male_c + donor,
  data = d1,
  family = binomial,
  chains = 4, 
  iter = 5000*2, 
  cores = 4, 
  seed = 84735)
```

### Priors, chain diagnostics, posterior predictive check
```{r}
prior_summary(m1b)

mcmc_trace(m1b, size = 0.1)
mcmc_dens_overlay(m1a)

summary(m1b)
```
- mcmc diagnostics all look good.

### List model variable names
```{r}
get_variables(m1b)
```

### Posterior predictive check
```{r}
pp_check(m1b, plotfun = "stat") +
    xlab("Frog survival rate")
```
- Posterior predictive distribution implied by regression model closely matches original data (based on mean frog survival).

### Calculate loo
```{r}
loo_m1b <- loo(m1b, cores = 4, save_psis = TRUE)
print(loo_m1b)
```

### Posterior analysis
```{r}
tidy(m1b, conf.int = TRUE, conf.level = 0.90)

mcmc_areas(m1b, area_method = "equal height", prob = 0.90) +
  geom_vline(xintercept = 0, color = "red", linetype = "dashed", lwd = 0.5) +  
  ggtitle("Model with standardized predictors only",
          "Posterior distributions with medians & 90% uncertainty intervals")

mean(bayes_R2(m1b))
```
- As expected, using standardized predictors allows easy comparison across predictors. Important predictors are unchanged. 
- Based on 90% uncertainty intervals, important predictors are snow_t1, elevation, first, donor70567, and donor72996.

## Compare models
```{r}
loo_compare(loo_m1a, loo_m1b)
```
- As expected, models with non-standardized and standardized predictor variables have identical predictive accuracy.  

## Possible grouping structures to add to model
The dataset has three groups: site_id, translocation_id, and donor site. Donor site is not suitable as a grouping variable because there are only 3 levels (i.e., site ids), not enough to make this useful. Site_id and translocation_id contain 12 and 24 levels, respectively, making them suitable as grouping variables. Including both site_id and translocation_id as nested grouping variables (i.e., translocations nested within site ids) also seems justified. Having more levels within a group would seem to increase the power to detect effects of predictors. With fewer levels/less variability, effect may be accounted for primarily by grouping variable itself instead of predictor. With only 1-4 translocations per site and only 12 site ids total, it is possible that model with site_id as sole grouping variable and model with site_id and translocation_id as nested grouping variables will attribute more of the variation in frog survival to the grouping variables and less to the predictors. A priori, that would suggest that the model with translocation_id as the grouping variable might be the most useful model structure (assuming that all models have similar posterior predictive accuracy). Lacking any strong rationale for  one model structure over another, built 3 models and compared their posterior predictive accuracy using loo. 

## Model using standardized predictors and site_id grouping variable
### Model specification
```{r}
m1c <- stan_glmer(
  surv ~ length_z + snowt_z + snowt1_z + day_z + bdload_z + elev_z + first_c + male_c + donor + (1 | site_id),
  data = d1,
  family = binomial,
  chains = 4, 
  iter = 5000*2, 
  cores = 4,
  seed = 84735)
```

### Priors & chain diagnostics
```{r}
prior_summary(m1c)

b1 <- c("(Intercept)", "length_z", "snowt_z", "snowt1_z", "day_z", "bdload_z", "elev_z", "first_c0.353", "male_c0.548", "donor70567", "donor72996")

mcmc_trace(m1c, size = 0.1, pars = b1)
mcmc_dens_overlay(m1c, pars = b1)

summary(m1c)
```

### Posterior predictive check
```{r}
pp_check(m1c, plotfun = "stat") +
    xlab("Frog survival rate")
```

### Calculate loo
```{r}
loo_m1c <- loo(m1c, cores = 4, save_psis = TRUE)
print(loo_m1c)
```

### Posterior analysis
```{r}
tidy(m1c, conf.int = TRUE, conf.level = 0.90)

mcmc_areas(m1c, area_method = "equal height", prob = 0.90, pars = b1) +
  geom_vline(xintercept = 0, color = "red", linetype = "dashed", lwd = 0.5) +  
  ggtitle("Model with standardized predictors & site_id as grouping variable",
          "Posterior distributions with medians & 90% uncertainty intervals")

mean(bayes_R2(m1c))

# Frog survival by site_id
mcmc_areas_ridges(m1c, regex_pars = "site_id") +
  geom_vline(xintercept = 0, color = "red", linetype = "dashed", lwd = 0.5) +  
  ggtitle("Frog survival for each recipient site_id")
```
- Based on 90% uncertainty intervals, important predictors reduced to elevation and male. 

## Model using standardized predictors and translocation_id grouping variable)
### Model specification
```{r}
m1d <- stan_glmer(
  surv ~ length_z + snowt_z + snowt1_z + day_z + bdload_z + elev_z + first_c + male_c + donor + (1 | translocation_id),
  data = d1,
  family = binomial,
  chains = 4, 
  iter = 5000*2, 
  cores = 4,
  seed = 84735)
```

### Priors & chain diagnostics
```{r}
prior_summary(m1d)

b1 <- c("(Intercept)", "length_z", "snowt_z", "snowt1_z", "day_z", "bdload_z", "elev_z", "first_c0.353", "male_c0.548", "donor70567", "donor72996")

mcmc_trace(m1d, size = 0.1, pars = b1)
mcmc_dens_overlay(m1d, pars = b1)

summary(m1d)
```

### Posterior predictive check
```{r}
pp_check(m1d, plotfun = "stat") +
    xlab("Frog survival rate")
```

### Calculate loo
```{r}
loo_m1d <- loo(m1d, cores = 4, save_psis = TRUE)
print(loo_m1d)
```

### Posterior analysis
```{r}
tidy(m1d, conf.int = TRUE, conf.level = 0.90)

mcmc_areas(m1d, area_method = "equal height", prob = 0.90, pars = b1) +
  geom_vline(xintercept = 0, color = "red", linetype = "dashed", lwd = 0.5) +  
  ggtitle("Model with standardized predictors & translocation_id as grouping variable",
          "Posterior distributions with medians & 90% uncertainty intervals")

mean(bayes_R2(m1d))

# Frog survival by translocation_id
mcmc_areas_ridges(m1d, regex_pars = "translocation_id") +
  geom_vline(xintercept = 0, color = "red", linetype = "dashed", lwd = 0.5) +  
  ggtitle("Frog survival for each translocation_id")
```
- Based on loo, model with translocation_id as grouping variable provides slightly better fit than model with site_id as grouping variable. 
- Based on 90% uncertainty intervals, important predictors are similar to those in model without grouping variable: snowt1_z, elev_z, male_c0.548, donor72996. 

## Model using standardized predictors and 2 grouping variables: translocation_id nested within site_id)
### Model specification
```{r}
m1e <- stan_glmer(
  surv ~ length_z + snowt_z + snowt1_z + day_z + bdload_z + elev_z + first_c + male_c + donor + (1 | translocation_id) + (1 | site_id),
  data = d1,
  family = binomial,
  chains = 4, 
  iter = 5000*2, 
  cores = 4,
  seed = 84735)
```

### Priors & chain diagnostics
```{r}
prior_summary(m1e)

b1 <- c("(Intercept)", "length_z", "snowt_z", "snowt1_z", "day_z", "bdload_z", "elev_z", "first_c0.353", "male_c0.548", "donor70567", "donor72996")

mcmc_trace(m1e, size = 0.1, pars = b1)
mcmc_dens_overlay(m1e, pars = b1)

summary(m1e)
```

### Posterior predictive check
```{r}
pp_check(m1e, plotfun = "stat") +
    xlab("Frog survival rate")
```

### Calculate loo
```{r}
loo_m1e <- loo(m1e, cores = 4, save_psis = TRUE)
print(loo_m1e)
```

### Posterior analysis
```{r}
tidy(m1e, conf.int = TRUE, conf.level = 0.90)

mcmc_areas(m1e, area_method = "equal height", prob = 0.90, pars = b1) +
  geom_vline(xintercept = 0, color = "red", linetype = "dashed", lwd = 0.5) +  
  ggtitle("Model with standardized predictors & translocation_id as grouping variable",
          "Posterior distributions with medians & 90% uncertainty intervals")

mean(bayes_R2(m1e))

# Frog survival by translocation_id
mcmc_areas_ridges(m1e, regex_pars = ("translocation_id")) +
  geom_vline(xintercept = 0, color = "red", linetype = "dashed", lwd = 0.5) +  
  ggtitle("Frog survival for each translocation_id")

# Frog survival by site_id
mcmc_areas_ridges(m1e, regex_pars = ("site_id")) +
  geom_vline(xintercept = 0, color = "red", linetype = "dashed", lwd = 0.5) +  
  ggtitle("Frog survival for each translocation_id")
```

## Compare 4 models with different grouping structures
```{r}
loo_compare(loo_m1b, loo_m1c, loo_m1d, loo_m1e)
```
- Based on expected log-predictive densities (ELPD), 3 models with grouping variables all have substantially higher posterior predictive accuracy than model without any grouping variables (elpd_diff of m1b is more than 2 SE lower than elpd_diff of m1c: elpd_diff = -18.2 $\pm$ 12.6, 2 SE interval does not overlap 0: -5.6 -- -30.8). 
- 3 models with 1 or more grouping variables are equivalent and have similar posterior predictive accuracy (elpd_diff of m1e and m1d are within 2 SE of elpd_diff of m1c).  
- m1d seems to provide more insight into the effect of the predictors, so chose this model for additional analysis. 

## Additional analysis of model m1d
### Posterior classification accuracy
```{r}
# Posterior predictive models for each frog in dataset
set.seed(84735)
survival_predict_1 <- posterior_predict(m1d, newdata = d1)

# Classify frog survival based on mean predicted survival probability
survival_predict_2 <- d1 %>% 
  mutate(survival_prob = colMeans(survival_predict_1),
         survival_class = as.numeric(survival_prob >= 0.5)) %>% 
  select(site_id, translocation_id, year, snowt1_z, elev_z, male_c, donor, surv, survival_prob, survival_class)

# Generate confusion matrix and calculate accuracy
survival_predict_2 %>% 
  tabyl(surv, survival_class) %>% 
  adorn_totals(c("row", "col"))

(471 + 138)/767 # overall_accuracy 
471/533 # true negative rate
138/234 # true positive rate

# Generate confusion matrix using function
set.seed(84735)
classification_summary(m1d, data = d1, cutoff = 0.5)
```
- Sensitivity or "true positive rate" measures the proportion of Y = 1 observations that are accurately classified.  
- Specificity or "true negative rate" measures the proportion of Y = 0 observations that are accurately classified (i.e., proportion of frogs correctly identified as not surviving).  

### Create reduced model containing only important predictors for subsequent analyses
- Reduced model is helpful because it reduces the number of values used to calculate predictive models. 
```{r}
m1d_reduce <- stan_glmer(
  surv ~ snowt1_z + elev_z + male_c + donor + (1 | translocation_id),
  data = d1,
  family = binomial,
  chains = 4, 
  iter = 5000*2, 
  cores = 4,
  seed = 84735)
```

### Confirm fit of reduced model is comparable to full model
```{r}
loo_m1d_reduce <- loo(m1d, cores = 4, save_psis = TRUE)
print(loo_m1d)

loo_compare(loo_m1d, loo_m1d_reduce)
```

### Simulate and plot posterior predictive models & observations
```{r}
m1d_group_means <- d1 %>% 
  group_by(translocation_id) %>% 
  summarize(count = n(), psurv = mean(surv), elev_z = mean(elev_z), snowt1_z = mean(snowt1_z)) %>% 
  mutate(
    male_c = as.factor(0.548), 
    donor = case_when(
      str_detect(translocation_id, "74976") | str_detect(translocation_id, "70556") ~ "70459",
      str_detect(translocation_id, "70413") ~ "70567",
      TRUE ~ "72996")) %>% 
  arrange(psurv)  # to arrange plot values/labels by psurv

predictions_m1d <- posterior_predict(m1d_reduce, newdata = m1d_group_means)

ppc_intervals(
  y = m1d_group_means$psurv,
  yrep = predictions_m1d, 
  prob = 0.5, 
  prob_outer = 0.8) +
ggplot2::scale_x_continuous(
  labels = m1d_group_means$translocation_id,
  breaks = 1:nrow(m1d_group_means)) +
  xaxis_text(angle = 90, vjust = 0.5) +
xlab("Translocation_id")
```
- Uncertainty intervals appear incorrect for binomial model, but not sure how to adjust.  
- Based on mean values, model does an adequate job of predicting survival based on group-level and individual-level characteristics.  
- What explains the low predicted survival at 70370 (both translocations), and high versus low predicted survival at 70550 for 1st and 2nd translocations, respectively? Both sites seem to contain high quality habitat (i.e., high elevation).  
- The predicted survival is based on assumption that only males are included. Run another set of models for females?  

### Plot posterior plausible relationships between each predictor and the probability of frog survival (conditional effects)
- As described in https://rdrr.io/cran/brms/man/conditional_effects.brmsfit.html, "When creating conditional_effects for a particular predictor (or interaction of two predictors), one has to choose the values of all other predictors to condition on. By default, the mean is used for continuous variables and the reference category is used for factors, but you may change these values via argument conditions. This also has an implication for the points argument: In the created plots, only those points will be shown that correspond to the factor levels actually used in the conditioning, in order not to create the false impression of bad model fit, where it is just due to conditioning on certain factor levels."  
- Using rstanarm, I was unable to do this as I expected to: Start with data = d1 and using "newdata = " to specify that predictions should be based on data in d1_elevz (for example), in which non-plotted continuous predictors are set to the mean value. Not sure if my alternative approach is correct.  
- Approach used: Instead of starting the routine with d1, I started with the new dataset in which all variables except the one of interest were set to mean values (continuous variables) and the reference level (categorical variables). This produced plots/results that were very similar to those produced in brms using the conditional_effects function. 

#### Effect of elevation
```{r}
d1_elevz <- d1 %>% 
  mutate(snowt1_z = mean(snowt1_z), 
         male_c = recode(male_c, "-0.452" = "0.548"),
         donor = recode(donor, "70567" = "70459", "72996" = "70459"))

d1_elevz %>% distinct(snowt1_z, male_c, donor) # ensure variables were set to correct values
  
d1_elevz %>% 
  select(site_id, translocation_id, snowt1_z, elev_z, male_c, donor, surv) %>% 
  add_epred_draws(m1d_reduce, ndraws = 100, re_formula = NA) %>% 
  ggplot(aes(x = elev_z, y = surv)) +
    geom_line(aes(y = .epred, group = .draw), color = "blue", size = 0.2, alpha = 0.5) +
#    geom_point(aes(y = .epred)) +
  labs(x = "Elevation of recipient site (standardized)", y = "Probability of 1-year frog survival")


# d1 %>%  # Test of add_predicted_draws used for similar purpose - unsuccessful
#   select(site_id, translocation_id, snowt1_z, elev_z, male_c, donor, surv) %>% 
#   add_predicted_draws(m1d_reduce, ndraws = 100, re_formula = NA, newdata = d1_elevz) %>% View()
#   ggplot(aes(x = elev_z, y = surv)) +
#     geom_line(aes(y = .epred, group = .draw), color = "blue", size = 0.2, alpha = 0.5) +
#     geom_point(aes(y = .epred)) +
#   labs(x = "Elevation of recipient site (standardized)", y = "Probability of 1-year frog survival")
```

#### Effect of winter severity in year t1
```{r}
d1_snowt1z <- d1 %>% 
  mutate(elev_z = mean(elev_z), 
         male_c = recode(male_c, "-0.452" = "0.548"),
         donor = recode(donor, "70567" = "70459", "72996" = "70459"))

d1_snowt1z %>% distinct(elev_z, male_c, donor) # ensure variables were set to correct values
  
d1_snowt1z %>% 
  select(site_id, translocation_id, snowt1_z, elev_z, male_c, donor, surv) %>% 
  add_epred_draws(m1d_reduce, ndraws = 100, re_formula = NA) %>% 
  ggplot(aes(x = snowt1_z, y = surv)) +
    geom_line(aes(y = .epred, group = .draw), color = "blue", size = 0.2, alpha = 0.5) +
#    geom_point(aes(y = .epred)) +
  labs(x = "Winter severity in year following translocation (standardized)", y = "Probability of 1-year frog survival") 
```

#### Effect of sex
```{r}
d1_malec <- d1 %>% 
  mutate(elev_z = mean(elev_z), 
         snowt1_z = mean(snowt1_z),
         donor = recode(donor, "70567" = "70459", "72996" = "70459"))

d1_malec %>% distinct(elev_z, snowt1_z, donor) # ensure variables were set to correct values
  
d1_malec %>% 
  select(site_id, translocation_id, snowt1_z, elev_z, male_c, donor, surv) %>% 
  add_epred_draws(m1d_reduce, ndraws = 100, re_formula = NA) %>% 
  ggplot(aes(x = male_c, y = surv)) +
    geom_boxplot(aes(y = .epred, x = male_c)) +
#    geom_point(aes(y = .epred)) +
    labs(x = "Sex (centered)", y = "Probability of 1-year frog survival")
```

#### Effect of donor
```{r}
d1_donor <- d1 %>% 
  mutate(elev_z = mean(elev_z), 
         snowt1_z = mean(snowt1_z),
         male_c = recode(male_c, "-0.452" = "0.548"))

d1_donor %>% distinct(elev_z, snowt1_z, male_c) # ensure variables were set to correct values
  
d1_donor %>% 
  select(site_id, translocation_id, snowt1_z, elev_z, male_c, donor, surv) %>% 
  add_epred_draws(m1d_reduce, ndraws = 100, re_formula = NA) %>% 
  ggplot(aes(x = donor, y = surv)) +
    geom_boxplot(aes(y = .epred, x = donor)) +
#    geom_point(aes(y = .epred)) +
    labs(x = "Donor population", y = "Probability of 1-year frog survival")
```

#### Joint effects of elevation and sex
```{r}
d1_elevz_malec <- d1 %>% 
  mutate(snowt1_z = mean(snowt1_z), 
         donor = recode(donor, "70567" = "70459", "72996" = "70459"))

d1_elevz_malec %>% distinct(snowt1_z, donor) # ensure variables were set to correct values
  
pal <- c("#0d0887", "#f89540")
d1_elevz_malec %>% 
  select(site_id, translocation_id, snowt1_z, elev_z, male_c, donor, surv) %>% 
  add_epred_draws(m1d_reduce, ndraws = 100, re_formula = NA) %>% 
  ggplot(aes(x = elev_z, y = surv, color = male_c)) +
    geom_line(aes(y = .epred, group = paste(male_c, .draw)), size = 0.4, alpha = 0.5) +
#    geom_point(aes(y = .epred)) +
    labs(x = "Elevation of recipient site (standardized)", y = "Probability of 1-year frog survival") +
    scale_color_manual(values = pal)  + 
    theme_classic()
```


## For comparison, fit model using brms and conditional_effects function to plot variable-specific effects
### Fit model
```{r}
m1d_reduce_brms <- brm(
  surv ~ snowt1_z + elev_z + male_c + donor + (1 | translocation_id),
  data = d1,
  family = bernoulli(),
  chains = 4, 
  iter = 5000*2, 
  cores = 4,
  seed = 84735)
```

### Plot conditional effects
```{r}
plot(conditional_effects(m1d_reduce_brms, spaghetti = TRUE, mean = TRUE, ndraws = 100), ask = FALSE)
```


## TO DO
- Validate priors, tune as necessary. Add final priors to model specification (in case the defaults change).  
- Check approach to plotting conditional effects.  
- Create analysis in which survival from first translocation to a site is a predictor of survival from subsequent translocations. For this analysis, will need to drop all sites at which only a single translocation was conducted.  
- Try un-standardizing predictor values in conditional-effects plots (and other plots?) to produce more intuitive visualizations.  








